Non-axisymmetric vertical velocity dispersion distributions produced ...

Download PDF
3 downloads 56 Views 3MB Size Report
Comment
Nov 20, 2016 - models. A vertically thin bar can lower σz along the bar and enhance it .... GA] 20 Nov 2016 ...... After reaching a quasi-steady state, the semi-.
Draft version November 22, 2016 Preprint typeset using LATEX style emulateapj v. 5/2/11

NON-AXISYMMETRIC VERTICAL VELOCITY DISPERSION DISTRIBUTIONS PRODUCED BY BARS Min Du1, 2 , Juntai Shen1, 2 , Victor P. Debattista2, 3

arXiv:1611.06489v1 [astro-ph.GA] 20 Nov 2016

Draft version November 22, 2016

ABSTRACT In barred galaxies, the contours of stellar velocity dispersions (σ) are generally expected to be oval and aligned with the orientation of bars. However, many double-barred (S2B) galaxies exhibit distinct σ peaks on the minor axis of inner bar, which we termed “σ-humps,” while two local σ minima are present close to the ends of inner bars, i.e., “σ-hollows.” Analysis of numerical simulations shows that σz -humps or hollows should play an important role in generating the observed σ-humps+hollows in low-inclination galaxies. In order to systematically investigate the properties of σz in barred galaxies, we apply the vertical Jeans equation to a group of well-designed three-dimensional bar+disk(+bulge) models. A vertically thin bar can lower σz along the bar and enhance it perpendicular to the bar, thus generating σz -humps+hollows. Such a result suggests that σz -humps+hollows can be generated by the purely dynamical response of stars in the presence of a, sufficiently massive, vertically thin bar, even without an outer bar. Using self-consistent N -body simulations, we verify the existence of vertically thin bars in the nuclear-barred and S2B models which generate prominent σ-humps+hollows. Thus the ubiquitous presence of σ-humps+hollows in S2Bs implies that inner bars are vertically thin. The addition of a bulge makes the σz -humps more ambiguous and thus tends to somewhat hide the σz -humps+hollows. We show that σz may be used as a kinematic diagnostic of stellar components that have different thickness, providing a direct perspective on the morphology and thickness of nearly face-on bars and bulges with integral-field unit spectroscopy. Subject headings: galaxies: kinematics and dynamics — galaxies: structure — galaxies: stellar content — galaxies: bulges 1. INTRODUCTION

Near-infrared imaging surveys have shown that bars are ubiquitous stellar structures; in the local Universe about two-thirds of disk galaxies host elongated stellar bars (Eskridge et al. 2000; Whyte et al. 2002; Laurikainen et al. 2004a; Men´endez-Delmestre et al. 2007; Marinova & Jogee 2007). The fraction is 0.25 − 0.3 if only strong bars are counted (e.g. Nilson 1973; de Vaucouleurs et al. 1991). From N -body simulations it is well known that bars can spontaneously form in galactic disks if the disk dynamical temperature (Toomre-Q) is not too high (e.g. Miller et al. 1970; Hohl 1971; Ostriker & Peebles 1973; Sellwood 1980, 1981; Athanassoula & Sellwood 1986a). Once formed bars are expected to be long-lived and difficult to destroy (Shen & Sellwood 2004; Debattista et al. 2006; Villa-Vargas et al. 2010; Athanassoula et al. 2013), which is supported by the fact that bars are typically composed of old stars (Gadotti & de Souza 2006; S´ anchez-Bl´ azquez et al. 2011). Observations of intermediate-redshift galaxies have revealed that the fraction of bars increases from ∼ 20% at z ∼ 0.84 to ∼ 65% in the local Universe (Sheth et al. 2008). As the frequency of violent interactions between galaxies decreases, the evolution of galaxies is driven mainly by internal processes, so-called secular evolution. Galactic bars are the most important driver of the secular evo1 Key Laboratory of Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China 2 Correspondence should be addressed to [email protected]; [email protected]; [email protected] 3 Jeremiah Horrocks Institute, University of Central Lancashire, Preston, PR1 2HE, UK

lution of disk galaxies (see the reviews of Kormendy & Kennicutt 2004; Kormendy 2013). By transferring angular momenta to the outer disk and dark matter halo, bars may grow longer and stronger, but rotate slower (Debattista & Sellwood 1998, 2000; Athanassoula 2003). Bars can drive the transport and accumulation of gas toward galactic central regions, thus triggering nuclear starbursts and, possibly, fuelling active galactic nuclei (AGN) (e.g. Shlosman et al. 1989, 1990; Buta & Combes 1996; Bournaud & Combes 2002; Maciejewski 2004a,b; Garc´ıa-Burillo et al. 2005; Hopkins & Quataert 2010; Kim et al. 2012; Emsellem et al. 2015; Li et al. 2015). Numerical simulations also suggest that bar formation can trigger the vertical buckling instability, leading to boxy/peanut (B/P) bulges (Raha et al. 1991; Merritt & Sellwood 1994). Being composed primarily of old stars, bars can be traced well in infrared bands where the dust extinction is much weaker than that in visible bands. The morphology of bars have been systematically investigated through ellipse fitting and Fourier decomposition of infrared images (e.g. Chapelon et al. 1999; Knapen et al. 2000; Laine et al. 2002; Laurikainen & Salo 2002; Laurikainen et al. 2002; Erwin 2005). Early dynamical studies of bars used long-slit spectroscopy of stars and ionized gas (e.g. Kuijken & Merrifield 1995; Bureau & Freeman 1999; Vega Beltr´an et al. 2001). In the last decade, the development of integral-field unit (IFU) spectroscopy has made it possible to study the 2D kinematics of nearby galaxies. The kinematics of disks and bars have been quantified in several IFU surveys, e.g. SAURON (de Zeeuw et al. 2002), ATLAS3D (Cappellari et al. 2011), CALIFA (S´anchez et al. 2012), DiskMass (Bershady et al. 2010),

2 and MaNGA (Bundy et al. 2015) (see review by Cappellari 2016). However, knowledge of the kinematic properties of bars is still incomplete. In early-type barred galaxies the central kinematic major axis is misaligned from the line of nodes (LON) by around ∼ 5◦ (Cappellari et al. 2007; Krajnovi´c et al. 2011). This is probably because the elongated streaming motions in bars distort the velocity fields, as shown in numerical studies (Miller & Smith 1979; Vauterin & Dejonghe 1997; Bureau & Athanassoula 2005). According to N -body simulations, the kinematic misalignment is not prominent in bars of early-type galaxies, possibly, because large random motions dominate (Du et al. 2016). Generally, faceon or moderately inclined bars are expected to generate oval velocity dispersion (σ) contours aligned with the bar (Debattista et al. 2005; Iannuzzi & Athanassoula 2015; Du et al. 2016). The most surprising σ features are the σ-humps and hollows found in double-barred (S2B) galaxies. Using SAURON IFU spectroscopy, de Lorenzo-C´ aceres et al. (2008) found that the σ maps of S2Bs reveal two local minima at the ends of inner bars, which they termed “σhollows.” Du et al. (2016) showed that such σ-hollows can be reproduced in the self-consistent simulations of S2Bs, which match well the σ features in the S2Bs in the ATLAS3D and SAURON surveys. The S2B simulations exhibit double-peaked σ enhancements along the minor axis of inner bars as well, termed “σ-humps,” which are also seen in the observations. Optical and near-infrared observations have revealed that multi-bar structures are quite common in the local Universe; almost one-third of early-type barred galaxies host S2B structures (Erwin & Sparke 2002; Laine et al. 2002; Erwin 2004). Many observations have shown that in S2Bs small-scale inner bars are dynamically decoupled from their large-scale outer counterparts (Buta & Crocker 1993; Friedli & Martinet 1993; Corsini et al. 2003). Inner bars are generally expected to rotate faster than outer ones. The physical origin of σ-humps+hollows is still unclear. Du et al. (2016) reported that σ-humps+hollows often accompanied nuclear bars in single-barred models. Therefore, σ-humps+hollows are not unique features of S2Bs, and cannot arise from the interaction of two bars. de Lorenzo-C´ aceres et al. (2008) proposed that σ-hollows are simply caused by the contrast of a dynamically cold inner bar embedded in a relatively hotter bulge. In Du et al. (2016) we compared the difference in intrinsic kinematics between the model with σ-humps+hollows to that without σ-humps+hollows. Their only difference is the double-peaked vertical velocity dispersion (σz ) enhancements perpendicular to the inner bar, i.e., σz humps, which must play an important role in generating σ-humps+hollows. In this paper, we investigate the σz properties in a family of analytical models of barred galaxies, i.e., bar+disk systems, which are introduced in Section 2. The analytical results are presented in Section 3, where we successfully explain the physical origin of σz -humps+hollows from a purely dynamical point of view. In Section 4, we test the effect of a massive bulge component on σz features. In Section 5, using the self-consistent nuclearbarred and large-scale single-barred simulations, we verify the analytical results. We further demonstrate that σz can be used as a kinematic diagnostic of the rela-

tive thickness of different stellar components. Finally, we summarize the conclusions of this work in Section 6. 2. METHOD 2.1. Vertical Jeans equation

Galactic disks are equilibrium systems whose stellar kinematics must satisfy the Jeans equations (Binney & Tremaine 2008, Equation 4.208) which were first applied to stellar systems by Jeans (1922). For disk galaxies, the Jeans equations generally cannot be uniquely solved in the disk plane (the x − y plane in Cartesian coordinates) without assumptions. In this paper we are only concerned with the vertical Jeans equation, i.e., the z direction, which can be written as 2 2 ) ∂(ρ? σz2 ) ∂(ρ? σxz ) ∂(ρ? σyz ∂Φ + + = −ρ? = ρ? Fz , (1) ∂x ∂y ∂z ∂z 2 where σiz = vi vz − v¯i v¯z = vi vz ; i = x, y, z, assuming 2 v¯z is always zero. σzz is written as σz2 for short. Fz is the vertical gravitational force from the total potential (Φ) which includes the contributions from stars and the dark matter halo. ρ? is the volume density of the stellar component. With the boundary condition ρ? = 0 as z → ∞, the integral from z to ∞ gives Z ∞ ρ? (z)σz2 (z) = − ρ? (z 0 )Fz (z 0 )dz 0  (2) Zz ∞  ∂(ρ? σxz ) ∂(ρ? σyz ) + dz 0 . + ∂x ∂y z

Corresponding to the anisotropic pressure forces, the second-order velocity moments, σxz and σyz , are omitted in the following analyses, thus Z ∞ ρ? (z)σz2 (z) ≈ − ρ? (z 0 )Fz (z 0 )dz 0 . (3) z

The legitimacy of this assumption in barred galaxies will be discussed in Section 5.1. The integral of the velocity dispersion in the face-on view is obtained by R +∞ R +∞ ρ σ 2 dz ρ? σz2 dz

2 −∞ ? z σz = R +∞ = −∞ Σ ρ dz −∞ ? (4) R +∞ R ∞ − −∞ z ρ? (z 0 )Fz (z 0 )dz 0 dz ≈ . Σ Therefore, the vertical dynamics give a simple relation between the density distribution of stars and the vertical velocity dispersion σz . Assuming a reasonable vertical density distribution, the axisymmetric form of this relation has been used as a kinematic estimator of the stellar disk mass in DiskMass IFU survey (Bershady et al. 2010, 2011; Martinsson et al. 2013a,b; Angus et al. 2015). In the DiskMass survey, the influence of bars is generally ignored by selecting a sample of unbarred or small/weakly barred galaxies. 2.2. Equilibrium bar+disk models A σz map for any arbitrary density distribution can be numerically computed using Eqn. 4. In order to study the σz features in barred galaxies, we use a family of bar+disk models that are embedded in a dark matter

3 TABLE 1 Settings of the bar+disk systems Name

hz

MB

Ferrers n

a/b/c

E0 E1 E2 E3 L1 L2 E4 E5 E6 E7 E8 E9 E10

0.3 0.3 0.3 0.3 0.1 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.3

0 0.01 0.03 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.20 0.20

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1.0/0.4/0.1 1.0/0.4/0.1 1.0/0.4/0.1 1.0/0.4/0.1 1.0/0.4/0.1 1.0/0.2/0.1 1.0/0.6/0.1 1.0/0.4/0.3 1.0/0.4/0.5 2.0/0.4/0.1 2.0/0.4/0.1 2.0/0.4/0.3

Note. — From left to right: model name, disk scaleheight hz , bar mass MB , Ferrers n, and axial ratio of bars a/b/c.

potential. In cylindrical coordinates, the disk density we use is given by a simple double-exponential distribution ρD (R, z) =

Σ0 R z exp(− − ), 2hz hR hz

(5)

where Σ0 , hR , and hz are the central surface density, scalelength, and scaleheight, respectively, of the disk component. Thus the disk mass MD is 2πΣ0 h2R . To simplify the following analyses, we use the same unit system as in Du et al. (2015), i.e., MD = G = hR = V0 = 1. The bar component is modelled as a triaxial Ferrers ellipsoid (Ferrers 1877)  ρB0 (1 − m2 )n m ≤ 1 ρB (x, y, z) = (6) 0 m > 1, where m2 = x2 /a2 + y 2 /b2 + z 2 /c2 in the Cartesian coordinate. The bar is aligned along the x-axis; the values of a, b, and c determine the semi-major axis, semi-minor axis, and thickness, respectively, of the bar. The axial ratio b/a corresponds to the ellipticity () of the bar. The Ferrers n parameter determines how fast the density decreases outward. Photometric observations show that the typical density profiles of bars are shallow (flat) and clearly truncated in early-type galaxies, while in latetype galaxies they tend to decrease outward following a more exponential profile (Elmegreen & Elmegreen 1985; Chapelon et al. 1999; Laine et al. 2002; Laurikainen & Salo 2002; Laurikainen et al. 2002; Kim et al. 2016). The central density of the bar, ρB0 , is numerically calculated for a given total mass of the bar, MB . We use the same dark matter potential as in the N -body simulations of Du et al. (2015) 1 2 V ln(r2 + rh2 ), (7) 2 h where Vh = 0.6 and rh = 15. In such bar+disk systems, there are six free parameters in total: the disk scaleheight hz , bar mass MB , and four structural parameters of the bar a, b, c, and Ferrers n. The Ferrers n is fixed at 1.0 to generate a shallow bar model. Given a set of parameters, we use a Nx × Ny × Nz = 201 × 201 × 401 grid to calculate the gravitational potential of the total system with a parallel 3D Poisson ΦDM =

solver PSPFFT (Budiardja & Cardall 2011). The model is located at the geometric center of the grid, and the spatial resolution is constant at 0.01 along the z direction as z ∈ [−2.0, 2.0]. Including the dark matter potential, we numerically compute σz using Eqn. 4. Systematic studies of near-infrared images of barred galaxies have shown that the semi-major axis of bars varies up to 2.5hR , with the mean values ∼ 1.3hR and ∼ 0.6hR for early-type and late-type galaxies, respectively (Erwin 2005; Men´endez-Delmestre et al. 2007; D´ıaz-Garc´ıa et al. 2016). The scalelength of disk galaxies seems to be independent of their Hubble type (de Jong 1996; Graham & de Blok 2001; Fathi et al. 2010). However, optical and near-infrared observations of edgeon galaxies find a decreasing trend of the scaleheight hz from early-type to late-type galaxies (de Grijs 1998; Schwarzkopf & Dettmar 2000; Bizyaev et al. 2014; Mosenkov et al. 2015). Table 1 shows the set of parameters we choose to study the σz features of barred galaxies. According to the empirical relation of de Grijs (1998), the ratio hR /hz varies from 1 − 5 (the median value ∼ 4) in early-type spirals (T ≤ 1) to 5 − 12 (the median value ∼ 9) in very latetype spirals (T ≥ 5). Thus we vary hz from 0.3 to 0.1 in our analytical models. In the cases of hz = 0.3, we name such early-type-like (“E”) double-exponential models as “E*.” The late-type-like double-exponential models are named as “L*” in the cases of hz = 0.1 or 0.2. The model E0 is a purely axisymmetric disk without a bar, i.e., MB = 0. A typical bar is used in most models by setting a = hR = 1.0 and b/a = 0.4. In the models E1-7 and L1-2 the bar length (a = 1.0) is half of that in E8-10 (a = 2.0). We truncate the disk at twice the bar length in order to reduce the calculation time, which allows us to obtain a high enough spatial resolution in the x − y plane (0.02 in E0-7 and L1-2; 0.04 in E8-10). We have confirmed that using a larger truncation radius makes negligible difference in the σz map. The dark matter potential also has a tiny effect on σz . To better understand the σz features, we further decompose the contributions of the bar and the disk components as follows

2 ΣD 2 ΣB 2 σz = σz D + σ , Σtot Σtot z B

(8)

where ΣD , ΣB , and Σtot are the surface density of the disk, bar, and total stellar system, respectively. hσz iD and hσz iB are the intrinsic vertical velocity dispersions of the

2 disk and

the bar, respectively. According to Eqn. 4, σz D and σz2 B can be calculated as R +∞ R ∞ − −∞ z ρD Fz dz 0 dz

2 σz D = ΣD (9) R +∞ R ∞ − −∞ z ρB Fz dz 0 dz

2 σz B = , ΣB where Fz is the total vertical force. In following analyses, we weight the disk and p the bar σz by the surface density, i.e., the disk σz = ΣD hσz2 iD /Σtot and the bar σz = p ΣB hσz2 iB /Σtot . Thus the disk and the bar σz are their respective contributions to the total σz . The bulge σz will be defined similarly when a bulge is added. As shown

4

Fig. 1.— Total surface density Σtot and σz maps of the models E1-3 in Table 1, showing the variation of σz fields with increasing bar mass. From top to bottom, the bar mass is set to 0.01MD (E1), 0.03MD (E2), and 0.05MD (E3), respectively. From left to right: Σtot , total σz , disk σz , and bar σz . The isodensity contours are equally separated in logarithmic space and σz contours of each map are overlaid using black dashed and white solid curves, respectively.

in Table 1, we explore the effect of bar mass MB (E0-3), ellipticity a/b (E3-5), thickness c (E3, E6-7), and semimajor axis a (E3, E8-10) of the Ferrers bar. E3 exhibits prominent σ-humps+hollows. The models L1-2 show the effect of disk scaleheight hz . 3. ANALYTICAL RESULTS: THE σZ FEATURES IN BAR+DISK SYSTEMS

3.1. Bar mass

Fig. 2.— 1D σz profiles along the minor (red) and major (blue) axes of the bars in the models E0-3, varying the bar mass MB . The black solid profile in the left panel shows the σz profile of the pure-disk model E0.

The pure-disk model, E0, generates an axisymmetric σz distribution as expected. Adding a bar and keeping its shape constant at a/b/c = 1.0/0.4/0.1, we increase the bar mass by varying MB from 0.01MD in E1 to 0.05MD in E3. The total surface density Σtot (the first column) and resulting σz maps of E1-3 are shown in Fig. 1. The second column shows the total σz of the bar+disk systems. The disk and the bar σz , weighted by the respective surface density, are shown in the third and fourth columns, respectively. As the bar mass increases, we can see more prominent σz -humps+hollows along the minor/major-axis of the bar. In order to better appreciate the amplitudes of the σz -humps+hollows, we plot the 1D σz profiles along the minor (red) and major (blue) axes of the bars (Fig. 2). For comparison, the σz of E0 is overlaid in black. As shown in Fig. 2, E0-3 exhibit almost the same distribution of σz at large radii (R > 1.2), suggesting that the influence of the bar is

5

Fig. 3.— Models E4, E3, and E5 in Table 1, showing the variation of σz fields when varying the minor-to-major axial ratio b/a to 0.2 (E4), 0.4 (E3), and 0.6 (E5).

axisymmetric bar potential. In contrast, σz values are reduced along the major axis of the bar, thus forming σz -hollows. As shown in Fig. 1, the disk σz maps (the third column) also show σz -humps+hollows as in the total σz maps, while none are present in the bar σz (the fourth column). The oval σz contours of the bar are aligned with the bar. Therefore, surprisingly, although the σz -humps are supported by the bar potential, they are mainly present in the disk component, extending beyond the bar along the minor axis. This result is consistent with observations (de Lorenzo-C´aceres et al. 2008; Du et al. 2016) and simulations (Du et al. 2016) of S2Bs. 3.2. Bar ellipticity and thickness

Fig. 4.— 1D σz profiles along the minor (red) and major (blue) axes of the bars in the models E4-3 and E5, varying the minor-tomajor axial ratio b/a of the bar.

important only in the inner region, where it dominates. With the bar mass increasing, in E2-3 the σz values are significantly enhanced on the minor axis of the bar, i.e., σz -humps form, which are clearly induced by the non-

In the analysis above, we have shown that even a relatively lightweight bar (0.05MD ) can generate prominent σz -humps+hollows. However, in most IFU observations and N -body simulations of barred galaxies, bars do not usually generate σz -humps+hollows. Currently, σ-humps+hollows have been seen only in the cases of S2Bs. In order to identify the condition for generating σz -humps+hollows, we study the effect of bar properties (a, b, and c) on σz . As shown in Table 1, fixing a = 1.0, we vary b and c in models E3-7. The bar ellipticity is varied from 0.8 (b/a = 0.2) in E4 to 0.4 (b/a = 0.6) in E5 (Fig. 3). The variation of the 1D profiles of σz -humps+hollows is shown in Fig. 4. Here the thickness of the bars is fixed at c = 0.1, i.e., a vertically thin bar. It is clear that a larger ellipticity, i.e., smaller

6

Fig. 5.— Models E3, E6, and E7 in Table 1, showing the variation of σz fields with varying bar thickness c from 0.1 in E3 to 0.5 in E7.

Fig. 7.— Total σz maps of the models L1-2. The disk scaleheight hz varies from 0.1 in L1 to 0.2 in L2. The overlaid black dashed and white solid curves show the isodensity and σz contours, respectively.

Fig. 6.— 1D σz profiles along the minor (red) and major (blue) axes of the bars in the models E3 and E6-7, varying the thickness c of the bar.

b/a, generates more prominent σz -humps as well as hollows. In models E3 and E6-7 we vary the bar thickness from 0.1 in E3 to 0.5 in E7 using a constant b/a = 0.4 (Fig. 5). As shown in Fig. 6, the central σz -drop gradually be-

comes a σz -peak as the bar thickness increases. Because of the enhancement of the bar σz (the fourth column of Fig. 5), the σz -humps+hollows become less prominent when the bar is thick. The disk σz is almost unchanged with increasing bar thickness (the third column). As a result, the total σz contours of E7 are oval and aligned with the bar. In Fig. 7 we show the total σz maps of the latetype-like models L1-2. Using a vertically thinner disk in L1 (hz = 0.1) and L2 (hz = 0.2), σz is reduced. There are no prominent σz -humps+hollows present in L1. Thus a vertically even thinner bar is required to generate σz -humps+hollows in late-type galaxies which are expected to be thinner than early-type galaxies.

7

Fig. 8.— Models E8-10 in Table 1, showing the σz maps of the long bar+disk models. The bar is twice the size of the bars in E1-7, i.e., a = 2.0, b = 0.8. The bar mass in E8-10 is 0.05MD , 0.2MD , and 0.2MD , respectively. The bars in E8-9 (c = 0.1) are vertically thinner than the one in E10 (c = 0.3).

the σz features in the long bar+disk models E8-10 using a = 2.0, b = 0.8 (Table 1). The bar in E8 has the same mass (0.05MD ) and thickness (c = 0.1) as E3, thus the size increase makes the bar potential shallower. As shown in the first row of Fig. 8, E8 generates quite round σz contours as the shallow bar potential supports only weak non-axisymmetric σz features. We set a more massive bar of mass 0.2MD in E9-10, varying c from 0.1 in E9 to 0.3 in E10. As shown in Fig. 8, there are no prominent σz -humps+hollows present in E9-10, although the moderately enhanced σz patterns are somewhat rectangular shaped in their outer parts. Such a result suggests that it is more difficult to generate central σz -humps+hollows in long bars than in short ones. Compared to E3, E9 has a similar bar σz distribution, but its disk σz is much larger at the center as a result of shallower potential. 3.4. Vertical density distribution of bars Fig. 9.— Map of the σz difference, obtained by subtracting the total σz field of E3 from that of the model using a vertically exponential density profile with a constant hBz = hz = 0.3. The overlaid black dashed and white solid curves show the isodensity and σz difference contours, respectively.

3.3. Bar length

Using the models E1-7, we have studied the conditions for generating σz -humps+hollows in galaxies hosting a typical bar of length a = 1.0. We further examine

According to the analyses above, σz -humps+hollows are primarily generated by the dynamical response of stars to the potential of a vertically thin, sufficiently massive, and relatively short bar. In photometric observations, we can easily measure the length, ellipticity, and mass of bars, especially in low-inclination galaxies, whereas we know little about their vertical density distributions. Although numerical simulations have been widely used to study the 3D morphology and orbital structure of bars (Pfenniger 1984; Martinet & de Zeeuw

8

Fig. 10.— Model E3B, which is identical to the E3 model but with the addition of a bulge of mass Mbulge = 0.30MD . In the left panel, we plot the logarithmic surface density profiles of the total system (Σtot ), bar (ΣB ), bulge (Σbulge ), and disk (ΣD ), where the red and the blue profiles correspond to the minor and the major axes, respectively. The middle panel shows the logarithmic vertical density distributions of the total system (ρtot ), bar (ρB ), bulge (ρbulge ), and disk (ρD ) at the center (x, y) = (0, 0). The right panel shows the total σz profiles along the minor (red) and major (blue) axes of the bar.

Fig. 11.— 2D Σtot and σz maps of E3B (Fig. 10).

qFig. 12.— Bulge σz , weighted by the bulge surface density, i.e., Σbulge hσz2 ibulge /Σtot , of the model E3B. The overlaid black dashed and white solid curves show the isodensity and σz contours, respectively, of the bulge component.

1988; Pfenniger & Friedli 1991; Sellwood & Wilkinson 1993; Skokos et al. 2002a,b; Patsis et al. 2003, 2002; Harsoula & Kalapotharakos 2009; Valluri et al. 2016), such theoretical models remain poorly tested by observations. We find that the bar σz is largely determined by the bar thickness, while the σz -humps arising in disks

Fig. 13.— Map of the σz difference, obtained by subtracting the σz field of the model E3B from that of the model using the constant scaleheight 0.3. The overlaid black dashed and white solid curves show the isodensity and σz difference contours, respectively.

seem to be insensitive to its properties. It suggests that σz can be used as a tracer of the bar thickness. However, our assumption of the vertical density distribution on bars is still questionable. The exponential (ρ(z) ∝ exp(−z/hBz )) and isothermal (ρ(z) ∝ sech2 (z/2hBz )) profiles have also been widely used to approximate the vertical density distribution of real bars for the purpose

9 of estimating the bar strength (Buta & Block 2001; Laurikainen & Salo 2002; Laurikainen et al. 2004a,b, 2005; Salo et al. 2015; D´ıaz-Garc´ıa et al. 2016). In this paper we considered only models using Ferrers bars. But we have verified that a much thinner bar is also required to generate prominent σz -humps+hollows for a typical bar with vertically exponential or isothermal density profiles. In Fig. 9 the σz difference is obtained by subtracting the total σz of E3 from that of the model using a vertically exponential bar whose scaleheight is the same as the disk, i.e., hBz = hz = 0.3. This helps to quantify how the variation of vertical density distribution of bars affects σz . As the difference in the total σz is almost only caused by the difference in the bar σz , the positive σz difference closely traces the thin bar in E3. The σz difference outside of the bar is close to zero. Thus the σz difference may be used as a diagnostic of the relative thickness of bars and their host disks. 4. THE EFFECT OF BULGES

In early-type disk galaxies, a large fraction of the luminosity comes from a massive spheroidal bulge. Having large random motions, bulges may affect the σz features significantly where σz -humps+hollows arise. In order to study the effect of a bulge on σz , we add an oblate, spheroidal power-law bulge in E3 using Eqn. 2.207 of Binney & Tremaine (2008) 2

2

ρbulge (R, z) = ρb0 m−αb e−m /rb (R ≤ rb ), (10) p where m = R2 + z 2 /qb2 ≥ 0.1. We set αb = 1.8 and qb = 0.6 (Binney & Merrifield 1998, Section 10.2.1). In order to avoid the singularity at the center, ρbulge is set to be constant at m ≤ 0.1. The bulge is truncated at rb = 1.5, so the bar is fully embedded in the bulge. The bulge mass is set to 0.3MD . This bar+disk+bulge system is named as E3B. Along the minor and major axes of the bar, the profiles of the surface density and σz are shown in the left and the right panels, respectively, of Fig. 10. The vertical density profiles at the center are shown in the middle panel. It can clearly be seen that the bulge is more massive than the bar at any position. The 2D Σtot and σz maps are shown in Fig. 11. As shown in the total σz map, the presence of the bulge significantly raise σz in the bulge region. It generates similar σz -humps along the minor axis to the simulations in Du et al. (2016), where the central σz -drop becomes more flat-topped (the right-most panel of Fig. 10) than E3. In the bulge (Fig. 12) the central σz contours are slightly oval and perpendicular to the bar. Such a result suggests that, in the bar potential the bulge component is not as responsive as the disk component, thus generating weaker non-axisymmetric σz features. The main influence of the bulge is to make the σz -humps less obvious by enhancing the central σz , thus hiding the σz -humps+hollows to a certain degree. By varying the thickness of the bar in such bar+disk+bulge systems (not shown here), we verify again that the bar needs to be much thinner than its host disk in order to generate visible σz -humps+hollows. As presented in Section 3.4, the positive σz difference can be used as a tracer of the thin bar. As shown in Fig. 13, we obtain the σz difference of E3B using the same approach. We firstly regenerate the vertical den-

sity distribution of the whole system with the exponential function using a constant scaleheight 0.3, which is used to recalculate σz . Then the original σz field is subtracted from the recalculated σz . Outside the bulge the σz difference is close to zero; the regions having positive and negative σz difference trace well the intrinsic face-on morphology of the thin bar and thick bulge, respectively. Debattista et al. (2005) showed that the fourth-order Gauss-Hermite moment h4 can be used as a kinematic diagnostic for bulges in nearly face-on galaxies. Here we show that σz differnce can be used as an alternative kinematic diagnostic of the stellar components having different thickness, e.g. thin bars and thick bulges, in barred galaxies. 5. N -BODY SIMULATIONS OF NUCLEAR AND LARGE-SCALE BARS

Using the bar+disk(+bulge) models, we have demonstrated that a vertically thin bar is required to generate σz -humps+hollows in barred galaxies. Such models allow us to study the effect of any single parameter by fixing the others. However, it is difficult to measure the 3D density distribution, especially perpendicular to the disk plane, in real galaxies. In order to verify the analytical results above, we study the 3D density distribution of self-consistent N -body simulations from Du et al. (2015, 2016). The unique advantage of simulations is that the 3D density distribution is completely known. As shown in Fig. 14, we have studied two representative cases, namely, a nuclear-barred simulation “NB” (the top row) and a large-scale single-barred simulation “SB” (the bottom row). Here we briefly summarize the properties of the NB and the SB models (see more details in Du et al. 2015). Starting from a pure exponential disk with 4 million particles, the models were evolved using a 3D cylindrical polar grid code, GALAXY (Sellwood & Valluri 1997; Sellwood 2014). The unit system of the simulations is the same as the analytical models in Section 3. By reducing the Toomre-Q in the inner region, the initial inner disk generally triggers a significant nuclear bar instability, forming nuclear-barred galaxy or double-barred galaxy (Fig. 1 in Du et al. 2015). Possibly because of the heating of spirals driven by the nuclear bar, the outer disk in the NB model becomes too hot to form a bar (Athanassoula & Sellwood 1986b; Du et al. 2015). After reaching a quasi-steady state, the semimajor axis of the nuclear bar extends to ∼ 0.7 of the initial hR , making it a quite short bar. The NB model exhibits similar σ-humps+hollows as the standard S2B (Du et al. 2015, 2016). Thus the outer bar is not a necessary condition for generating σ-humps+hollows. It is worth emphasizing that the initial thickness of the NB model is smoothly lowered to 0.05 inside R < 1.0 from 0.1 in the outer region. In this case the NB model generates more prominent σ-humps+hollows (the top-left panel of Fig. 14) than the cases of using a radially constant thickness of 0.1. Using a dynamically hotter initial inner disk normally leads to a large-scale single bar. The SB model here is exactly the same model as in Fig. 6 of Du et al. (2016) where the σz contours are oval and aligned with the bar (the bottom-left panel of Fig. 14). The semi-major axis of the bar in the SB model is ∼ 3.0. Both the NB and the SB models are thickened in their inner regions (R ∼ 1.5),

10

Fig. 14.— Maps of the simulated σz (left) and the analytical σz (middle) calculated using Eqn. 4, and the residual σz (right) of the nuclear-barred (NB, top) and the large-scale single-barred (SB, bottom) models. The residual σz , obtained by subtracting the analytical σz from the simulated σz , corresponds to the contribution of the anisotropic pressure in the simulations. We fix the color-bars of the left and middle columns. A much smaller range is used in the right column, as the maximum residual σz is at ∼ 5 − 10% level of the simulated σz . The surface density contours are overlaid in black, separated in equal intervals in logarithmic space.

where boxy/peanut (B/P) bulges possibly form as seen from the edge-on view. Thus in the NB model the nuclear bar is embedded in the host bulge. 5.1. Quantifying the uncertainty due to the anisotropic

pressure in barred galaxies In this study we have assumed the velocity crossterms are unimportant in the vertical dynamics (see Section 2.1), which is generally considered to be a good approximation in axisymmetric systems. However, this assumption is not obviously justified for non-axisymmetric bars which induce large streaming motions, possibly causing a systematic error in the σz calculation. In order to quantify the anisotropic pressure caused by the velocity cross-terms, we apply the vertical kinematic estimation to the NB and SB models. The density distribution and associated vertical force from the simulations are used to calculate the analytical σz (the middle column of Fig. 14). By subtracting the analytical σz from the simulation’s actual σz (the left column), we obtain the residual σz (the right column) that corresponds to the contribution of the anisotropic pressure. We have confirmed that the residual σz here is well consistent with the numerically calculated anisotropic pressure. The residual σz is roughly equal to zero all over the disk for the SB model. Only in the very central region is the residual σz positive at ∼ 10% level, which has no effect on σz -humps+hollows. In the NB model there is an extensive positive residual σz (∼ 5% level) along the minor axis of the bar, which is possibly related to the elongated streaming motions in the nuclear bar. We have checked

the standard S2B model as well, in which the anisotropic pressure enhances σz values along the minor axis of the inner bar at a similarly low level as the NB model. The maximum ellipticity of the simulated bars here reaches ∼ 0.6. In observations some late-type bars can be very narrow and strong, in which case the importance of the anisotropic pressure may increase. Therefore, a cautious conclusion is that the anisotropic pressure is negligible in galaxies containing a normal or weak bar. 5.2. A kinematic diagnostic of vertical thickness: σz

As the anisotropic pressure is negligible, the σz features of the NB and SB models are mainly determined by their 3D density distributions and associated potentials. In this section, we investigate whether a vertically thin bar exists in the NB model, which our analysis in Section 3 suggests is a necessary condition for generating σz -humps+hollows (Fig. 15 and 16). The SB model is shown for comparison purpose. In order to reduce the noise, we select the painted regions (annuli and filled dots in the left column of Fig. 15) to average the vertical density distribution. In both the NB (top) and SB (bottom) models the filled dots correspond to the minor axis (green), major axis (magenta), and center (black) of their bars. The cyan (at R = 1.0) and the red (at R = 3.0) annuli represent the B/P bulge and the outer disk regions, respectively. The red outer disk annulus in the NB model is not shown, as the computed region at R = 3.0 is beyond the boundary [−2.0, 2.0] of the image. In Fig. 16 the average density distributions of each of these regions are indicated by

11

Fig. 15.— Numerically calculated σz using different vertical density distributions and their σz difference (right) maps. Based on Eqn. 4, the σz maps in the left panels are calculated using the original 3D density distributions from the NB (top) and SB (bottom) simulations. In the middle panels, without changing the surface density, we recalculate the σz maps by using a vertically exponential profile with a constant scaleheight. The constant scaleheights are set to the linear-fitting of the vertical density distributions of the NB (hz = 0.21) and SB (hz = 0.24) outer disks (Fig. 16), respectively. The right panels show the σz difference between the σz calculated using the original density and that using a vertically exponential density distribution. The smoothed surface density and σz contours are overlaid in black and white, respectively. In the left panels, the painted dots and annuli mark the regions we use to average the vertical density profiles in Fig. 16. In the NB model the red annulus at R = 3.0 is beyond the boundary of the image in the NB model, thus not shown here.

Fig. 16.— Average density (ln¯ ρ) profiles of the NB (left) and the SB (right) models along the z direction. The profiles correspond to the vertical density distributions of the center (black), minor (green), and major (magenta) axes of the bar, B/P bulge (cyan), and outer disk (red) regions. The average regions are marked with the filled dots and annuli in the same color in Fig. 15. The dashed profiles represent the extrapolated linear-fitting of the ln¯ ρ profiles at the outer disk (red) and the center (black) and minor axis (green) of the bar. The fitted scaleheights are given in the legend.

12 the solid profiles using the same color. The dashed profiles represent the extrapolated linear-fitting of the ln¯ ρ profiles at each region. In the bar regions we only fit the region close to the mid-plane where the bar should dominate (for the NB z ∈ [−0.3, 0.3], while for the SB z ∈ [−0.5, 0.5]). The density profiles on the major axis of the bar (magenta) and the B/P bulge (cyan) cannot be fitted by a linear relation. The fitted scaleheight values at each region are given in the legend. The nuclear bar of the NB model (hz = 0.11 at the minor axis area) is vertically thinner than the host disk (hz = 0.21), which agrees well with the analytical expectation for generating σz -humps+hollows. In contrast, the bar of the SB model is as thick as the disk, except for its very central region; thus it exhibits no σz -humps+hollows. We calculate the σz difference using the same approach as in Sections 3.4 and 4. In the left panels of Fig. 15 we show the numerically calculated σz using the original 3D density distributions of the NB and SB models. In the middle panels σz is recalculated using the vertically exponential profiles with a constant scaleheight. The scaleheights used here are set to the linear-fitting results of the outer disks (NB hz = 0.21; SB hz = 0.24). Then the σz difference maps (the right panels) are obtained by subtracting the σz maps in the left panels from those in the middle panels. In this case, the non-zero σz difference represents the difference of vertical thickness from the outer disk. The σz difference is roughly equal to zero in the outer disk. In the NB model the σz difference is qualitatively consistent with the bar+disk+bulge model (Fig. 13). In the thin bar region the σz difference is positive at ∼ 10 − 20% level, while in the thick B/P bulge region it turns to be negative (∼ 15 − 30% level). As visually confirmed from the edge-on view, the NB model hosts a nearly boxy bulge of radius R ∼ 1.5. The negative σz difference traces the face-on morphology of such a boxy bulge. In the bottom-right panel, the B/P bulge of the SB model should correspond to the peanut-shaped negative region (∼ 25% level) in the σz difference map. A positive σz difference only appears at the very central region (marked with the filled black dot in Fig. 15) where ρ¯ is peaked around the mid-plane. In conclusion, the σz difference seems to be a good kinematic diagnostic for the stellar components having different thickness, e.g. thick bulges and thin bars. It may shed new light on the 3D geometry of bars and bulges in the face-on views of barred galaxies. It is worth emphasizing that, for real galaxies, hz is generally estimated either from the empirical relation in de Grijs (1998) or the observed σz in the outer disk by assuming a reasonable mass-to-light ratio. In practice, the estimation of hz still has a large uncertainty, and the mass-tolight ratio is not constant. This may cause large errors in the estimation of surface density. The practicality of this method will be tested in future work. 6. SUMMARY

By applying the vertical Jeans equation to a group of well-designed bar+disk(+bulge) models, we have systematically investigated the σz properties of barred galaxies from a purely dynamical point of view. The main conclusions can be summarized as: (1) Bars

can

dynamically

induce

signifi-

cant non-axisymmetric σz features, either σz -humps+hollows or oval σz contours aligned with bars. The properties of σz features are tightly related with the properties of bars, i.e., mass, length, ellipticity, and thickness. Generally, thick or long bar are more likely to generate oval σz contours aligned with bars. (2) We found that vertically thin bars can not only reduce σz along the major axis of bars but also enhance σz along the minor axis, thus generating σz -humps+hollows. Such σz -humps+hollows can explain the σ-humps+hollows appearing in the kinematic observations of double-barred galaxies. (3) As a dynamical response of stars to the potential of bars, the amplitude of σz -humps is proportional to the mass and ellipticity of bars, while it is almost independent of the bar thickness. σz -humps are mainly present in host disks, thus extending beyond bars. A thin bar mainly reduces σz in the bar region, thus generating σz -hollows. (4) We showed that σz -humps+hollows are preferentially found in galaxies harboring a short bar, e.g. inner bars of double-barred galaxies and single nuclear bars. σ-humps+hollows have been commonly observed in double-barred galaxies, while their frequency in nuclear-barred galaxies is still unclear. In long bar cases σz -humps+hollowsare less frequent, possibly because volume expansion makes bar potential shallower. (5) Using the bar+disk+bulge models, we show that the primary effect of a thick bulge is to make the σz humps weaker by enhancing the central σz . Thus σ-humps+hollows should not be explained by the contrast of dynamically cold bars and hot bulges as proposed in previous analysis. In IFU observations, an increasing number of σ-humps+hollows features have been identified in nearby S2Bs (de Lorenzo-C´aceres et al. 2008, 2013; Du et al. 2016). Du et al. (2016) presented self-consistent S2B simulations which match the kinematic observations of S2Bs. In this paper, we demonstrate that the existence of a vertically thin bar in the nuclear-barred simulation (NB) generates such σ-humps+hollows in small-scale (nuclear) bars. The interaction of multiple bars should play a minor role in generating σ-humps+hollows. The ubiquitous presence of σ-humps+hollows in S2Bs indicates that inner bars are vertically thin structures. Thus it suggests that inner bars are either not prone to thickening or they are younger structures formed in dynamically cold nuclear disks. However, the detailed stellar population analysis of S2Bs showed that inner bars are not young structures (de Lorenzo-C´aceres et al. 2012, 2013). In our simulations vertically thin bars also last for more than 5Gyr. Thus we propose that inner bars are weakly thickened after forming in initial nuclear disks. As embedded in galactic central regions, the vertical thickness of bars is rarely measured in real galaxies. In low-inclination cases, it is also very difficult to identify the morphology of bulges. An implication of this work is that σz may trace the stellar components having different thickness, e.g. thin bars and thick bulges. It may

13 provide a novel perspective on the 3D geometry of bars and bulges from IFU surveys for nearly face-on galaxies. M.D. thanks the Jeremiah Horrocks Institute of the University of Central Lancashire for their hospitality during a three month visit while this paper was in progress. M.D. thanks Adriana de Lorenzo-C´ aceres for constructive comments on the manuscript. The research presented here is partially supported by the 973 Program of China under grant no. 2014CB845700, by the National Natural Science Foundation of China under grant

nos.11333003, 11322326, and by the Strategic Priority Research Program “The Emergence of Cosmological Structures” (no. XDB09000000) of the Chinese Academy of Sciences. We acknowledge support from a Newton Advanced Fellowship #NA150272 awarded by the Royal Society and the Newton Fund. This work made use of the facilities of the Center for High Performance Computing at Shanghai Astronomical Observatory. V.P.D. is supported by STFC Consolidated grant # ST/J001341/1. V.P.D. was also partially supported by the Chinese Academy of Sciences President’s International Fellowship Initiative Grant (No. 2015VMB004).

REFERENCES Angus, R., Aigrain, S., Foreman-Mackey, D., & McQuillan, A. 2015, MNRAS, 450, 1787 Athanassoula, E. 2003, MNRAS, 341, 1179 Athanassoula, E., Machado, R. E. G., & Rodionov, S. A. 2013, MNRAS, 429, 1949 Athanassoula, E., & Sellwood, J. A. 1986a, MNRAS, 221, 213 —. 1986b, MNRAS, 221, 213 Bershady, M. A., Martinsson, T. P. K., Verheijen, M. A. W., et al. 2011, ApJ, 739, L47 Bershady, M. A., Verheijen, M. A. W., Swaters, R. A., et al. 2010, ApJ, 716, 198 Binney, J., & Merrifield, M. 1998, Galactic Astronomy Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition (Princeton University Press) Bizyaev, D. V., Kautsch, S. J., Mosenkov, A. V., et al. 2014, ApJ, 787, 24 Bournaud, F., & Combes, F. 2002, A&A, 392, 83 Budiardja, R. D., & Cardall, C. Y. 2011, Computer Physics Communications, 182, 2265 Bundy, K., Bershady, M. A., Law, D. R., et al. 2015, ApJ, 798, 7 Bureau, M., & Athanassoula, E. 2005, ApJ, 626, 159 Bureau, M., & Freeman, K. C. 1999, AJ, 118, 126 Buta, R., & Block, D. L. 2001, ApJ, 550, 243 Buta, R., & Combes, F. 1996, Fund. Cosmic Phys., 17, 95 Buta, R., & Crocker, D. A. 1993, AJ, 105, 1344 Cappellari, M. 2016, ArXiv e-prints, arXiv:1602.04267 Cappellari, M., Emsellem, E., Bacon, R., et al. 2007, MNRAS, 379, 418 Cappellari, M., Emsellem, E., Krajnovi´ c, D., et al. 2011, MNRAS, 413, 813 Chapelon, S., Contini, T., & Davoust, E. 1999, A&A, 345, 81 Corsini, E. M., Debattista, V. P., & Aguerri, J. A. L. 2003, ApJ, 599, L29 de Grijs, R. 1998, MNRAS, 299, 595 de Jong, R. S. 1996, A&A, 313, 45 de Lorenzo-C´ aceres, A., Falc´ on-Barroso, J., & Vazdekis, A. 2013, MNRAS, 431, 2397 de Lorenzo-C´ aceres, A., Falc´ on-Barroso, J., Vazdekis, A., & Mart´ınez-Valpuesta, I. 2008, ApJ, 684, L83 de Lorenzo-C´ aceres, A., Vazdekis, A., Aguerri, J. A. L., Corsini, E. M., & Debattista, V. P. 2012, MNRAS, 420, 1092 de Vaucouleurs, G., de Vaucouleurs, A., Corwin, Jr., H. G., et al. 1991, Third Reference Catalogue of Bright Galaxies. Volume I: Explanations and references. Volume II: Data for galaxies between 0h and 12h . Volume III: Data for galaxies between 12h and 24h . de Zeeuw, P. T., Bureau, M., Emsellem, E., et al. 2002, MNRAS, 329, 513 Debattista, V. P., Carollo, C. M., Mayer, L., & Moore, B. 2005, ApJ, 628, 678 Debattista, V. P., Mayer, L., Carollo, C. M., et al. 2006, ApJ, 645, 209 Debattista, V. P., & Sellwood, J. A. 1998, ApJ, 493, L5 —. 2000, ApJ, 543, 704 D´ıaz-Garc´ıa, S., Salo, H., Laurikainen, E., & Herrera-Endoqui, M. 2016, A&A, 587, A160 Du, M., Debattista, V. P., Shen, J., & Cappellari, M. 2016, ApJ, 828, 14 Du, M., Shen, J., & Debattista, V. P. 2015, ApJ, 804, 139 Elmegreen, B. G., & Elmegreen, D. M. 1985, ApJ, 288, 438

Emsellem, E., Renaud, F., Bournaud, F., et al. 2015, MNRAS, 446, 2468 Erwin, P. 2004, A&A, 415, 941 —. 2005, MNRAS, 364, 283 Erwin, P., & Sparke, L. S. 2002, AJ, 124, 65 Eskridge, P. B., Frogel, J. A., Pogge, R. W., et al. 2000, AJ, 119, 536 Fathi, K., Allen, M., Boch, T., Hatziminaoglou, E., & Peletier, R. F. 2010, MNRAS, 406, 1595 Ferrers, N. M. 1877, Q. J. Pure Appl. Math., 14 Friedli, D., & Martinet, L. 1993, A&A, 277, 27 Gadotti, D. A., & de Souza, R. E. 2006, ApJS, 163, 270 Garc´ıa-Burillo, S., Combes, F., Schinnerer, E., Boone, F., & Hunt, L. K. 2005, A&A, 441, 1011 Graham, A. W., & de Blok, W. J. G. 2001, ApJ, 556, 177 Harsoula, M., & Kalapotharakos, C. 2009, MNRAS, 394, 1605 Hohl, F. 1971, ApJ, 168, 343 Hopkins, P. F., & Quataert, E. 2010, MNRAS, 407, 1529 Iannuzzi, F., & Athanassoula, E. 2015, MNRAS, 450, 2514 Jeans, J. H. 1922, MNRAS, 82, 132 Kim, T., Gadotti, D. A., Athanassoula, E., et al. 2016, MNRAS, 462, 3430 Kim, W.-T., Seo, W.-Y., & Kim, Y. 2012, ApJ, 758, 14 Knapen, J. H., Shlosman, I., & Peletier, R. F. 2000, ApJ, 529, 93 Kormendy, J. 2013, Secular Evolution in Disk Galaxies, ed. J. Falc´ on-Barroso & J. H. Knapen, 1 Kormendy, J., & Kennicutt, Jr., R. C. 2004, ARA&A, 42, 603 Krajnovi´ c, D., Emsellem, E., Cappellari, M., et al. 2011, MNRAS, 414, 2923 Kuijken, K., & Merrifield, M. R. 1995, ApJ, 443, L13 Laine, S., Shlosman, I., Knapen, J. H., & Peletier, R. F. 2002, ApJ, 567, 97 Laurikainen, E., & Salo, H. 2002, MNRAS, 337, 1118 Laurikainen, E., Salo, H., & Buta, R. 2004a, ApJ, 607, 103 —. 2005, MNRAS, 362, 1319 Laurikainen, E., Salo, H., Buta, R., & Vasylyev, S. 2004b, MNRAS, 355, 1251 Laurikainen, E., Salo, H., & Rautiainen, P. 2002, MNRAS, 331, 880 Li, Z., Shen, J., & Kim, W.-T. 2015, ApJ, 806, 150 Maciejewski, W. 2004a, MNRAS, 354, 883 —. 2004b, MNRAS, 354, 892 Marinova, I., & Jogee, S. 2007, ApJ, 659, 1176 Martinet, L., & de Zeeuw, T. 1988, A&A, 206, 269 Martinsson, T. P. K., Verheijen, M. A. W., Westfall, K. B., et al. 2013a, A&A, 557, A131 —. 2013b, A&A, 557, A130 Men´ endez-Delmestre, K., Sheth, K., Schinnerer, E., Jarrett, T. H., & Scoville, N. Z. 2007, ApJ, 657, 790 Merritt, D., & Sellwood, J. A. 1994, ApJ, 425, 551 Miller, R. H., Prendergast, K. H., & Quirk, W. J. 1970, ApJ, 161, 903 Miller, R. H., & Smith, B. F. 1979, ApJ, 227, 785 Mosenkov, A. V., Sotnikova, N. Y., Reshetnikov, V. P., Bizyaev, D. V., & Kautsch, S. J. 2015, MNRAS, 451, 2376 Nilson, P. 1973, Uppsala general catalogue of galaxies Ostriker, J. P., & Peebles, P. J. E. 1973, ApJ, 186, 467 Patsis, P. A., Skokos, C., & Athanassoula, E. 2002, MNRAS, 337, 578 —. 2003, MNRAS, 342, 69

14 Pfenniger, D. 1984, A&A, 134, 373 Pfenniger, D., & Friedli, D. 1991, A&A, 252, 75 Raha, N., Sellwood, J. A., James, R. A., & Kahn, F. D. 1991, Nature, 352, 411 Salo, H., Laurikainen, E., Laine, J., et al. 2015, ApJS, 219, 4 S´ anchez, S. F., Kennicutt, R. C., Gil de Paz, A., et al. 2012, A&A, 538, A8 S´ anchez-Bl´ azquez, P., Ocvirk, P., Gibson, B. K., P´ erez, I., & Peletier, R. F. 2011, MNRAS, 415, 709 Schwarzkopf, U., & Dettmar, R.-J. 2000, A&A, 361, 451 Sellwood, J. A. 1980, A&A, 89, 296 —. 1981, A&A, 99, 362 —. 2014, ArXiv e-prints, arXiv:1406.6606 Sellwood, J. A., & Valluri, M. 1997, MNRAS, 287, 124 Sellwood, J. A., & Wilkinson, A. 1993, Reports on Progress in Physics, 56, 173 Shen, J., & Sellwood, J. A. 2004, ApJ, 604, 614 Sheth, K., Elmegreen, D. M., Elmegreen, B. G., et al. 2008, ApJ, 675, 1141

Shlosman, I., Begelman, M. C., & Frank, J. 1990, Nature, 345, 679 Shlosman, I., Frank, J., & Begelman, M. C. 1989, Nature, 338, 45 Skokos, C., Patsis, P. A., & Athanassoula, E. 2002a, MNRAS, 333, 847 —. 2002b, MNRAS, 333, 861 Valluri, M., Shen, J., Abbott, C., & Debattista, V. P. 2016, ApJ, 818, 141 Vauterin, P., & Dejonghe, H. 1997, MNRAS, 286, 812 Vega Beltr´ an, J. C., Pizzella, A., Corsini, E. M., et al. 2001, A&A, 374, 394 Villa-Vargas, J., Shlosman, I., & Heller, C. 2010, ApJ, 719, 1470 Whyte, L. F., Abraham, R. G., Merrifield, M. R., et al. 2002, MNRAS, 336, 1281